Quartic Integral Polynomial Pell Equations

TL;DR

This paper classifies all monic quartic polynomials with integer coefficients for which the polynomial Pell equation admits non-trivial polynomial solutions, advancing understanding of polynomial Pell equations in algebraic number theory.

Contribution

It provides a complete classification of monic quartic polynomials over integers that allow polynomial solutions to the Pell equation, a problem previously unresolved.

Findings

Complete classification of such quartic polynomials.

Identification of conditions for existence of polynomial solutions.

Extension of Pell equation theory to quartic polynomials.

Abstract

In this paper we classify all monic, quartic, polynomials $d(x)\in\mathbb{Z}[x]$ for which the Pell equation $$f(x)^2-d(x)g(x)^2=1$$ has a non-trivial solution with $f(x),g(x)\in\mathbb{Z}[x]$.